Elliptic problems with radiation conditions on edges of the boundary

S. A. Nazarov   ; B. A. Plamenevskii

Sbornik. Mathematics, Tome 77 (1994) no. 1, pp. 149-176

Résumé

A study is made of formulations of elliptic boundary value problems connected with the addition of radiation conditions on edges of the piecewise smooth boundary $\partial G$ of a domain $G\subset\mathbb{R}^n$. Such formulations lead to Fredholm operators acting in suitable function spaces with weighted norms. The basic means of description is the generalized Green formula, which contains in addition to the usual boundary integrals also integrals over an edge $M$ of bilinear expressions formed by the coefficients of the asymptotics of the solutions near $M$. Thus, the edge and the $(n-1)$-dimensional smooth part of the boundary are on the same footing-both $M$ and $\partial G\setminus M$ are represented by their contributions to the generalized Green formula. This permits the construction of a theory of elliptic problems in which the generalized Green formula takes the role of the usual Green formula in the smooth situation.

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